Question 4 - A-Level Maths

Edexcel FP2 — Question 4

Exam Board	Edexcel
Module	FP2 (Further Pure Mathematics 2)
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Polar coordinates
Type	Find constant from given area
Difficulty	Standard +0.8 This is a Further Maths polar coordinates question requiring knowledge of the area formula (½∫r²dθ), expansion of (a+3cosθ)², integration of cos²θ using double angle formulas, and solving for a constant. While systematic, it involves multiple non-trivial steps and is from FP2, making it moderately challenging but still a standard polar area problem.
Spec	4.09c Area enclosed: by polar curve

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a7ef3811-3594-4ecd-a616-36f42d26489b-06_428_803_233_577} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve with polar equation $$r = a + 3 \cos \theta , \quad a > 0 , \quad 0 \leqslant \theta < 2 \pi$$ The area enclosed by the curve is $\frac { 107 } { 2 } \pi$.
Find the value of $a$.

Show mark scheme Show mark scheme source

Question 4:

Part (a)

Answer	Marks	Guidance
Assume true for $n=k$: $z^k = r^k(\cos k\theta + \mathrm{i}\sin k\theta)$
$n=k+1$: $z^{k+1} = (z^k \times z =) r^k(\cos k\theta + \mathrm{i}\sin k\theta) \times r(\cos\theta + \mathrm{i}\sin\theta)$	M1	For using the result for $n=k$ to write $z^{k+1}$
$= r^{k+1}(\cos k\theta\cos\theta - \sin k\theta\sin\theta + \mathrm{i}(\sin k\theta\cos\theta + \cos k\theta\sin\theta))$	M1	For multiplying out and collecting real and imaginary parts using $\mathrm{i}^2=-1$, OR using sum of arguments and product of moduli
$= r^{k+1}(\cos(k+1)\theta + \mathrm{i}\sin(k+1)\theta)$	M1dep A1 cso	For using addition formulae to obtain single cos and sin terms; dependent on second M mark. Only award if all previous steps fully correct
$k=1$: $z^1 = r^1(\cos\theta + \mathrm{i}\sin\theta)$; True for $n=1$ $\therefore$ true for all $n$	A1 cso	All 5 underlined statements must be seen (5)

Alternative (Euler's form):

Answer	Marks	Guidance
$z = r(\cos\theta + \mathrm{i}\sin\theta) = re^{\mathrm{i}\theta}$	M1	May not be seen explicitly
$z^{k+1} = z^k \times z = (re^{\mathrm{i}\theta})^k \times re^{\mathrm{i}\theta} = r^k e^{\mathrm{i}k\theta} \times re^{\mathrm{i}\theta}$	M1
$= r^{k+1}e^{\mathrm{i}(k+1)\theta}$	M1dep on 2nd M
$= r^{k+1}(\cos(k+1)\theta + \mathrm{i}\sin(k+1)\theta)$	A1 cso
$k=1$: $z^1 = r^1(\cos\theta + \mathrm{i}\sin\theta)$; True for $n=1$ $\therefore$ true for all $n$	A1 cso	All 5 underlined statements must be seen

Part (b)

Answer	Marks	Guidance
$w = 3\left(\cos\frac{3\pi}{4} + \mathrm{i}\sin\frac{3\pi}{4}\right)$
$w^5 = 3^5\left(\cos\frac{15\pi}{4} + \mathrm{i}\sin\frac{15\pi}{4}\right)$	M1	For attempting to apply de Moivre to $w$ or attempting to expand $w^5$ and collecting real and imaginary parts
$w^5 = 243\left(\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}}\mathrm{i}\right) \left[= \frac{243\sqrt{2}}{2} - \frac{243\sqrt{2}}{2}\mathrm{i}\right]$	A1 cao	oe e.g. $3^5$ instead of 243 (2) [7]

## Question 4:

### Part (a)
| Assume true for $n=k$: $z^k = r^k(\cos k\theta + \mathrm{i}\sin k\theta)$ | | |
|---|---|---|
| $n=k+1$: $z^{k+1} = (z^k \times z =) r^k(\cos k\theta + \mathrm{i}\sin k\theta) \times r(\cos\theta + \mathrm{i}\sin\theta)$ | M1 | For using the result for $n=k$ to write $z^{k+1}$ |
| $= r^{k+1}(\cos k\theta\cos\theta - \sin k\theta\sin\theta + \mathrm{i}(\sin k\theta\cos\theta + \cos k\theta\sin\theta))$ | M1 | For multiplying out and collecting real and imaginary parts using $\mathrm{i}^2=-1$, OR using sum of arguments and product of moduli |
| $= r^{k+1}(\cos(k+1)\theta + \mathrm{i}\sin(k+1)\theta)$ | M1dep A1 cso | For using addition formulae to obtain single cos and sin terms; dependent on second M mark. Only award if all previous steps fully correct |
| $k=1$: $z^1 = r^1(\cos\theta + \mathrm{i}\sin\theta)$; True for $n=1$ $\therefore$ true for all $n$ | A1 cso | All 5 underlined statements must be seen **(5)** |

**Alternative (Euler's form):**
| $z = r(\cos\theta + \mathrm{i}\sin\theta) = re^{\mathrm{i}\theta}$ | M1 | May not be seen explicitly |
| $z^{k+1} = z^k \times z = (re^{\mathrm{i}\theta})^k \times re^{\mathrm{i}\theta} = r^k e^{\mathrm{i}k\theta} \times re^{\mathrm{i}\theta}$ | M1 | |
| $= r^{k+1}e^{\mathrm{i}(k+1)\theta}$ | M1dep on 2nd M | |
| $= r^{k+1}(\cos(k+1)\theta + \mathrm{i}\sin(k+1)\theta)$ | A1 cso | |
| $k=1$: $z^1 = r^1(\cos\theta + \mathrm{i}\sin\theta)$; True for $n=1$ $\therefore$ true for all $n$ | A1 cso | All 5 underlined statements must be seen |

### Part (b)
| $w = 3\left(\cos\frac{3\pi}{4} + \mathrm{i}\sin\frac{3\pi}{4}\right)$ | | |
|---|---|---|
| $w^5 = 3^5\left(\cos\frac{15\pi}{4} + \mathrm{i}\sin\frac{15\pi}{4}\right)$ | M1 | For attempting to apply de Moivre to $w$ or attempting to expand $w^5$ and collecting real and imaginary parts |
| $w^5 = 243\left(\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}}\mathrm{i}\right) \left[= \frac{243\sqrt{2}}{2} - \frac{243\sqrt{2}}{2}\mathrm{i}\right]$ | A1 cao | oe e.g. $3^5$ instead of 243 **(2) [7]** |

Show LaTeX source

4.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{a7ef3811-3594-4ecd-a616-36f42d26489b-06_428_803_233_577}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}

Figure 1 shows a sketch of the curve with polar equation

$$r = a + 3 \cos \theta , \quad a > 0 , \quad 0 \leqslant \theta < 2 \pi$$

The area enclosed by the curve is $\frac { 107 } { 2 } \pi$.\\
Find the value of $a$.\\

\hfill \mbox{\textit{Edexcel FP2  Q4}}

This paper (1 questions)

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Answer	Marks	Guidance
Assume true for \(n=k\): \(z^k = r^k(\cos k\theta + \mathrm{i}\sin k\theta)\)
\(n=k+1\): \(z^{k+1} = (z^k \times z =) r^k(\cos k\theta + \mathrm{i}\sin k\theta) \times r(\cos\theta + \mathrm{i}\sin\theta)\)	M1	For using the result for \(n=k\) to write \(z^{k+1}\)
\(= r^{k+1}(\cos k\theta\cos\theta - \sin k\theta\sin\theta + \mathrm{i}(\sin k\theta\cos\theta + \cos k\theta\sin\theta))\)	M1	For multiplying out and collecting real and imaginary parts using \(\mathrm{i}^2=-1\), OR using sum of arguments and product of moduli
\(= r^{k+1}(\cos(k+1)\theta + \mathrm{i}\sin(k+1)\theta)\)	M1dep A1 cso	For using addition formulae to obtain single cos and sin terms; dependent on second M mark. Only award if all previous steps fully correct
\(k=1\): \(z^1 = r^1(\cos\theta + \mathrm{i}\sin\theta)\); True for \(n=1\) \(\therefore\) true for all \(n\)	A1 cso	All 5 underlined statements must be seen (5)

Answer	Marks	Guidance
\(z = r(\cos\theta + \mathrm{i}\sin\theta) = re^{\mathrm{i}\theta}\)	M1	May not be seen explicitly
\(z^{k+1} = z^k \times z = (re^{\mathrm{i}\theta})^k \times re^{\mathrm{i}\theta} = r^k e^{\mathrm{i}k\theta} \times re^{\mathrm{i}\theta}\)	M1
\(= r^{k+1}e^{\mathrm{i}(k+1)\theta}\)	M1dep on 2nd M
\(= r^{k+1}(\cos(k+1)\theta + \mathrm{i}\sin(k+1)\theta)\)	A1 cso
\(k=1\): \(z^1 = r^1(\cos\theta + \mathrm{i}\sin\theta)\); True for \(n=1\) \(\therefore\) true for all \(n\)	A1 cso	All 5 underlined statements must be seen

Answer	Marks	Guidance
\(w = 3\left(\cos\frac{3\pi}{4} + \mathrm{i}\sin\frac{3\pi}{4}\right)\)
\(w^5 = 3^5\left(\cos\frac{15\pi}{4} + \mathrm{i}\sin\frac{15\pi}{4}\right)\)	M1	For attempting to apply de Moivre to \(w\) or attempting to expand \(w^5\) and collecting real and imaginary parts
\(w^5 = 243\left(\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}}\mathrm{i}\right) \left[= \frac{243\sqrt{2}}{2} - \frac{243\sqrt{2}}{2}\mathrm{i}\right]\)	A1 cao	oe e.g. \(3^5\) instead of 243 (2) [7]